A196670 The Chebyshev primes of index 4.

5, 7, 17, 19, 31, 37, 41, 43, 53, 59, 67, 73, 79, 83, 101, 103, 107, 127, 131, 149, 157, 163, 179, 181, 197, 199, 211, 223, 227, 257, 269, 277, 281, 317, 331, 337, 347, 353, 379, 389, 419, 421, 439, 461, 463, 467, 479, 491, 499, 509, 541, 563, 569, 577, 617

Offset

1,1

Comments

The sequence consists of such odd prime numbers p that satisfy li(psi(p^4)) - li(psi(p^4-1)) < 1/4, where li(x) is the logarithmic integral and psi(x) is the Chebyshev psi function.

Links

Dana Jacobsen, Table of n, a(n) for n = 1..75

M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT], 2011.

Maple

# The function PlanatSole(n, r) is in A196667.
A196670 := n -> PlanatSole(n, 4); # Peter Luschny, Oct 23 2011

Mathematica

ChebyshevPsi[n_] := Log[LCM @@ Range[n]];
Reap[Do[If[LogIntegral[ChebyshevPsi[p^4]] - LogIntegral[ChebyshevPsi[p^4 - 1]] < 1/4, Print[p]; Sow[p]], {p, Prime[Range[2, 120]]}]][[2, 1]] (* Jean-François Alcover, Jul 14 2018, updated Dec 06 2018 *)

Prog

(Magma)
Mangoldt:=function(n);
if #Factorization(n) eq 1 then return Log(Factorization(n)[1][1]); else return 0; end if;
end function;
tcheb:=function(n);
x:=0;
for i in [1..n] do
x:=x+Mangoldt(i);
end for;
return(x);
end function;
jump4:=function(n);
x:=LogIntegral(tcheb(NthPrime(n)^4))-LogIntegral(tcheb(NthPrime(n)^4-1));
return x;
end function;
Set4:=[];
for i in [2..1000] do
if jump4(i)-1/4 lt 0 then Set4:=Append(Set4, NthPrime(i)); NthPrime(i); end if;
end for;
Set4;
(Sage)
def A196670(n) : return PlanatSole(n, 4)
# The function PlanatSole(n, r) is in A196667.
# Peter Luschny, Oct 23 2011
(Perl) use ntheory ":all"; forprimes { say if 4 *(LogarithmicIntegral(chebyshev_psi($_**4)) - LogarithmicIntegral(chebyshev_psi($_**4-1))) < 1 } 3, 100; # Dana Jacobsen, Dec 29 2015

Crossrefs

Cf. A196667, A196668, A196669.

Sequence in context: A106120 A106121 A185022 * A075304 A168245 A242929

Adjacent sequences: A196667 A196668 A196669 * A196671 A196672 A196673

Keyword

nonn

Author

Michel Planat, Oct 05 2011

Extensions

More terms from Dana Jacobsen, Dec 29 2015

Status

approved